Agile software development is a conceptual framework for undertaking software engineering projects. There are a number of agile software development methods, such as those espoused by The Agile Alliance. Most agile methods attempt to minimize risk by developing software in short timeboxes, called iterations, which typically last one to four weeks. Each iteration is like a miniature software project of its own, and includes all of the tasks necessary to release the mini-increment of new functionality: planning, requirements analysis, design, coding, testing, and documentation. While an iteration may not add enough functionality to warrant releasing the product, an agile software project intends to be capable of releasing new software at the end of every iteration. At the end of each iteration, the team reevaluates project priorities.
The advantages of agile software development are that you begin seeing working (though not necessarily comlpete) software very quickly. The idea is to get developing and do the entire project in a series of short iterations with a functioning app at the end of each iteration. In each iteration, new features are added until the app is complete, or at least in a releasable state. The downsides are that with a sizeable team of people - 3+ say - communication as to which way things are developing can be problematic if the developers don't communicate clearly. If the developers are in-experienced they will tend to get lost as the more experienced and capable people leave them behind.